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ELSEVIER
o
Remote Sensing
O(
Environment
Remote Sensing of Environment84 (2003) 561-571
www.elsevier.com/Iocate/rse
An improved strategy for regression of biophysical Variables
and Landsat ETM+ data
Warren B. Cohen a'*, Thomas K. Maiersperger b, Stith T. Gower c, David P. Turner b
aForest;y Sciences Laboratory, Pacific Northwest Research Station. USDA Forest Service. 3200 SW Jefferson Win; Corvallis, OR 97331. USA
b Department of Forest Science. ForestO, Sciences Laborato~ 020. Oregon State University. Corvallis. OR 97331. USA
~Department of Forest Ecology and Management. 1630 Linden Drive, University of Wisconsin. Madison. WI 53706, USA
Received 5 December 2001; accepted 15 July 2002
Abstract
Empirical models are important tools for relating field-measured biophysical variables to remote sensing data. Regression analysis has
been a popular empirical method of linking these two types of data to provide continuous estimates for variables such as biomass, percent
woody canopy cover, and leaf area index (LAI). Traditional methods of regression are not sufficient when resulting biophysical surfaces
derived from remote sensing are subsequently used to drive ecosystem process models. Most regression analyses in remote sensing rely on a
single spectral vegetation index (SVI) based on red and near-infrared reflectance from a single date of imagery. There are compelling reasons
for utilizing greater spectral dimensionality, and for including SVIs from multiple dates in a regression analysis. Moreover, when including
multiple SVIs and/or dates, it is useful to integrate these into a single index for regression modeling. Selection of an appropriate regression
model, use of multiple SVIs from multiple dates of imagery as predictor variables, and employment of canonical correlation analysis (CCA)
to integrate these multiple indices into a single index represent a significant strategic improvement over existing uses of regression analysis in
remote sensing.
To demonstrate this improved strategy, we compared three different types of regression models to predict LAI for an agro-ecosystem and
live tree canopy cover for a needleleaf evergreen boreal forest: traditional ( Yon X) ordinary least squares (OLS) regression, inverse (X on Y)
OLS regression, and an orthogonal regression method called reduced major axis (RMA). Each model incorporated multiple SVIs from
multiple dates and CCA was used to integrate these. For a given dataset, the three regression-modeling approaches produced identical
coefficients of determination and intercepts, but different slopes, giving rise to divergent predictive characteristics. The traditional approach
yielded the lowest root mean square error (RMSE), but the variance in the predictions was lower than the variance in the observed dataset.
The inverse method had the highest RMSE and the variance was inflated relative to the variance of the observed dataset. RMA provided an
intermediate set of predictions in terms of the RMSE, and the variance in the observations was preserved in the predictions. These results are
predictable from regression theory, but that theory has been essentially ignored within the discipline of remote sensing.
© 2002 Elsevier Science Inc. All rights reserved.
Keywords: Regressionanalysis; Biophysical variables; Landsat ETM+
1. I n t r o d u c t i o n
Biogeochemical cycling models are increasingly run in a
spatially explicit mode, requiring as model drivers moderate
to high spatial resolution surfaces of land cover and leaf area
index (LAI) derived from satellite imagery (Bonan, 1993;
Reich, Turner, & Bolslad, 1999: Running, Batdocchi, Turner,
Gower, Bakwin, & Hibbard. 1999). Mapping of continuous
variables like LAI from high-resolution imagery such as
* Corresponding author. Tel.: +1-541-750-7322; fax: +1-541-7587760.
E-mail address: warren.cohen@orst.edu(W.B. Cohen).
Landsat TM or ETM+ has largely depended on modeling
empirical relationships derived from single-date spectral
vegetation indices (SVIs). The most important of these are
the normalized difference vegetation index (NDVI) and its
counterpart, the simple ratio (SR) (Chen & Cihlar, 1996;
Fassnacht, Gower, MacKenzie, Nordheim, & Lillesand,
1997; White, Running, Nemani. Keane, & Ryan, 1997).
These and other ratio-based indices, although important,
utilize only a fraction of the spectral information available
in many image datasets (Cohen, Spies. & Fiorella, 1995).
Moreover, with the cost of ETM+ data substantially reduced
from that of its predecessor, TM, there are increasing opportunities to utilize multiple dates of imagery in these analyses.
0034-4257/02/$ - see front matter © 2002 Elsevier Science Inc. All rights reserved.
PII: S0034-4257(02)00173-6
562
~B. Cohenet al. / Remote Sensing of Environment 84 (2003) 561-571
Traditional methods for empirical modeling of continuous variables, such as LAI, from SVIs rely on ordinary least
squares (OLS) regression (Steel & Tome. 1980), a technique that has important limitations for such applications
(Curran & Hay, 1986). In particular, a violation of assumptions about measurement error can have undesirable effects
on OLS estimates of the biophysical variable. In spite of the
cogent arguments against the use of OLS regression in
remote sensing offered by Cun'an and Hay (I 986), we could
find only one subsequent remote sensing paper that heeded
their advice (Larsson, 1993). Alternative regression models
may provide improved estimates of biophysical variables in
remote sensing. Several such models are discussed in the
literature, but almost all of that literature is outside of our
discipline.
When conducting regression analyses that utilize multiple SVIs and multi-date data, it would be useful to construct
a single, integrated index to represent the multiple predictor
variables. This would facilitate visual assessment of model
strength and whether the integrated relationship is linear. An
integrated index could also help in subsequent analyses, or
for screen viewing and interpretation (similar to the NDVI
or SR). Additionally, an integrated index would be useful
for comparisons among possible model fornmlations. Most
important, however, is that certain regression procedures are
best conducted in a simple linear context, and thus rely on a
single predictor variable. These needs can be met using a
statistical tool known as canonical correlation analysis
(CCA).
~
1.1. Objective
The goal of this paper is to demonstrate an improved
strategy for regression modeling of biophysical variables in
remote sensing. That strategy includes use of multiple SVIs
from multiple dates of ETM+ imagery, development of a
CCA-based index that integrates these, and choice of an
appropriate type of regression model. We test three regression-modeling approaches: traditional OLS (RegT), inverse
OLS (Reg0, and reduced major axis (RMA). The test is
done for two biophysical variables, one in each of two
different biomes, to highlight the general applicability of the
analyses and results. At an agricultural site, we model LAI
for two separate dates; at a boreal forest site, live tree cover
is modeled. The objective is to compare and contrast the
three regression approaches in terms of basic statistical
characteristics of the predicted variables relative to the
statistical characteristics of the observed variables. Numerous examples of such comparisons exist in the general
literature, and the lessons learned are applied in various
disciplines. However, the two papers in remote sensing
literature that address this issue (Curran & Hay, 1986;
Larsson, 1993) have been essentially ignored. With continued use of regression in remote sensing, and an increased
reliance on Landsat imagery to drive ecosystem process
models with regression-derived surfaces, it is imperative
that we consider the weaknesses of our common methods
and the potential strengths of alternative methods.
1.2. Background
1.2.1. Spectral vegetation indices (SVIs) and related linear
combinations
The value of SVIs for modeling the relationship between
vegetation variables and reflectance data is well established.
In particular, since their inception, the SR (Birth & McVey,
1968) and the NDVI (Rouse. Haas, Schell, & Deermg,
1974) have dominated the remote sensing and related
literature (e.g., Chen & Cihlar. 1996; Huete, Jackson, &
Post. 1985; Sellers, 1987; Tucke~; 1979; Turner, Cohen.
Kennedy, Fassnacht, Briggs, 1999). Modifications to these
indices have been proposed to account for background
effects associated with incomplete canopy cover (Huete,
1988), some of which take advantage of shortwave-infrared
reflectance (Brown, Chen, Leblanc, & Cihlar, 2000; Nemani. Pierce, Running, & Band, 1993).
Although these "ratio-based" indices have the advantage
of being simple to understand and apply, an alternative set
of indices, called "n-space indices" (Jackson, 1983), is
designed to more fully exploit the spectral domain of
reflectance data. Numerous such indices exist, including
the Perpendicular Vegetation Index (Richardson & Wiegand, 1977) and the widely used Tasseled Cap, which
consists of the brightness, greenness (Kauth & "thomas.
1976), and wetness (Crist & Cicone, 1984) indices. The
Tasseled Cap indices, in particular, provide standardized
coefficients for all spectral bands of Landsat MSS and TM
data. One study, across a forested scene containing bare
ground, brush, and broadleaf and needleleaf forests of
varying ages, demonstrated that brightness, greenness, and
wetness accounted for 85% of the total spectral variability
contained in a single date of TM reflectance data (Cohen et
al., t995); this, in comparison to only 52% in the red and
near-infrared bands.
The temporal domain of spectral data can greatly enhance our ability to map vegetation (Helmer, Brown, &
Cohen, 2000; Lefsky, Cohen, & Spies, 2001; Loveland et
al., 2000; Oetter, Cohen. Berterretche, Maiersperger, &
Kennedy. 2001). Incorporating a temporal series of data
into SVIs directly, however, has been given minimal attention. Malila (1980) described the change magnitude and
angle calculations, which are indices of a sort, required for
change vector analysis (CVA). Whereas CVA was designed
for two spectral dimensions and two dates of imagery, it has
been crudely extended to three spectral dimensions (Virag &
Colwetl, 1987), and the magnitude calculation has been
generalized to n-dimensions by Lambin and Strahler (1994).
The concept for generalizing CVA angles and magnitudes
for n spectral and/or temporal dimensions was described
by Cohen and Fiorella (1998), but they stopped short
of implementing the procedure. Collins and Woodcock
(1996) developed a two-date Tasseled Cap transformation
W.B. Cohen et al. /Remote Sensing of Environment 84 (2003) 561-571
fur use in change detection, but an n-date transformation
was not attempted.
Principal components analysis (PCA) is an attractive
means of incorporating spectral data from numerous dates
into a small set of axes that contain most of the spectral
infornaation contained in the full multispectral, multitemporal dataset (Eastman & Fulk. 1993: Richards, 1984).
Although not vegetation indices per se, the value of PCA
for reducing the size of the spectral-temporal dataset is
great. However, there are two important problems using
PCA for spectral-temporal analyses. First, the resulting
axes are dataset-dependent. Although this means that it is
difficult to generalize the interpretation of PCA axes to other
datasets, this is not unique to PCA, as the same problem is
common to all correlation-based, empirical analyses. The
second and more meaningful problem is that the coefficients
for PCA axes are normally obtained without regard for the
axes' relationships with the variable we are interested in
predicting (e.g., LAI).
1.2.2. Regression and related analysis
In the simple linear case, OLS regression analysis is an
empirical approach for modeling the relationship between
two observed variables, X and Y. The form of the OLS
regression model is
Y = [~o + fl~x + ~,
(l)
where Yis the variable to be predicted, Xis the variable Yis
predicted from, //0 is the intercept, ~ is the slope of the
relationship between X and Y, and e is error. Data for the
analysis are supplied by paired observations of the two
variables. Commonly, one variable is difficult or costly to
measure (e.g., vegetation attributes from field sampling),
and the other is relatively easy or inexpensive to observe
(e.g., SVIs from remote sensing). Although often the intent
of OLS regression is to determine the feasibility of estimating or predicting the expensive variable from observations
of the inexpensive one, sometimes the analysis is used
simply to determine the form and strength of the relationship between the two variables. If the objective is the latter,
it is not particularly important which variable is Xand which
is Y, and common in the remote sensing literature are both X
and Y representing the vegetation variable and the SVI
(Butera, 1986: Chen & Cihlar, 1996; Cohen, 1991: While
el al., 1997). If we are interested in actually using the
regression model to predict one variable from the other,
however, the distinction between X and Y becomes very
important. This is because of specifications and assumptions
associated with OLS regression.
One specification is that Y is the dependent variable and
X is the independent variable (.Steel & Tome, 1980).
Although it can be argued that spectral response is dependent on vegetation state and not the other way around, much
of the remote sensing literature reports the vegetation
attribute being modeled as the dependent variable. Cun'an
and l tay (1986) discuss this as the "specification problem".
563
Specification in this manner is important because of an
assumption associated with OLS regression: that the independent variaNe, X(e.g., an SVI), is measured without error
(Steel & Ton'ie, 1980). As the coefficients for the regression
equation are calculated by minimizing the sums of squares
of error in Y (e.g., LAI), illustrated graphically by CreTan
and Hay (1986), the result is an attenuation (or compression)
of the variance of LAI predictions. In other words, values
above the mean of Y tend to be underpredicted and values
below .the mean tend to be overpredicted (e.g., Cohen,
Maiersperger. Spies. & Oetter. 2001 ; Hudak, Lef~ky, Cohen,
& Berterretche, 2002).
An alternative form of OLS regression is inverse
estimation (Brown, 1979), also known as inverse regression (Cohen, 1991) and'calibration (Scheff~. 1973). Curran
and Hay (1986.1 refer to this as X on Y regression and
illustrate the concept graphically. With inverse estimation,
the specification problem is addressed in that the dependent and independent variables are properly assigned, or
"specified" (e.g., X is the vegetation variable and Y is the
SVI). In practice then, to predict the vegetation variable,
the coefficients for the OLS regression model are derived
using Eq. (1) and then the equation must be inverted to
solve for X, such that X=( Y - [Jo)/fll. The error term in Eq.
(1), e., is expressed as prediction residuals for each observation.
Although for remote sensing, inverse estimation eliminates the specification problem, it does not address the more
important problem that Xis assumed to be measured without
error. Ctm'an and Hay (1986) provide an in-depth discussion
of sources of error for both X and Y in remote sensing. The
impact of measurement error in X when using inverse
estimation is known to be the opposite to that of its effect
using the Yon X form of OLS regression, i.e., amplification
of the variance of predicted biophysical values such that
values above the mean of X are overestimated and those
below the mean are underestimated.
Recognizing that there are errors in both Xand Y, Cun'an
and Hay (1986) tested three alternative methods to predict
grassland LAI from the SR: Wald's grouping method, RMA,
and an alternative least squares procedure that incorporates a
priori knowledge of relative errors in X and g They
recommended using Wald's method or RMA if no estimates
of measurement error are available, and the alternative least
squares procedure if such estimates are available. From a
practical perspective, it will be rare for analysts to have
precise estimates of error from all the various sources
associated with measurements of vegetation and spectral
variables. As such, it is perhaps more prudent to make no
assumptions regarding the relative amounts of measurement
error and use RMA or Wald's method. In spite of the
convincing arguments made by Curran and Hay (1986),
we could find only one subsequent remote sensing article
that used one of these methods (Larsson, 1993), where
RMA was used to predict woodland canopy cover from
single-date NDVI measurements.
564
W.B. Cohen et al. /Remote Sensing of Environment 84 (2003) 561-571
RMA is one of a class of similar models known as
orthogonal regression, total least squares regression, or
errors-in-variables modeling, depending on the discipline
inwhich the specific technique was developed (Van Huffel,
1997). Orthogonal regression minimizes the sum of squared
orthogonal distances from measurement points to the model
function. The RMA version of orthogonal regression is
graphically depicted in Curran and Hay (1986). Van Hufl~'l
(1997) contains examples of orthogonal regression's usage
in astronomy, meteorology, 3-D motion estimation, biomedical signal processing, and multivariate calibration. RMA,
specifically, is quite commonly applied in allometry (Conrad & Gutmann, 1996; Gower, Kucharik, & Norman, 1999;
Nicol & Mackauer, 1999; Niklas & Buckman, 1994).
Besides making no assumptions about errors in X and Y,,
RMA likewise makes no assumptions about dependency.
C;onrad and Gutmann (1996) refer to RMA as geometric
mean regression, in that the slope (/30 is defined as the ratio
of sample standard deviation for Yover the sample standard
deviation for AT, thus preserving in the model the relative
variance structure of the sample dataset. The effect of this is
to minimize or eliminate any attenuation or amplification of
predictions. For RMA, flo is defined as the sample mean of
Y minus the quantity fl~ times the sample mean of X. One
important component of the slope term (rio is that it must be
given the sign ( + / - ) of the correlation between X and Y
(Conrad & Gutmann. 1996), which is not given by Curran
and Hay (11986) or Larsson (1993). The form of the
regression model is identical to Eq. (1), but the calculations
of rio and fl~ are different. Mathematical similarities in the
formulations of the two OLS and the RMA regression
models mean that the model intercepts are all equivalent,
as are the coefficients of determination. What differ among
these models are the root mean square errors (RMSEs) and
the slopes of the relationships.
1.2.3. Canonical correlation analysis (CCA)
OLS regression has both simple (single X) and multiple
(several X) forms (Steel & Tome, 1980). The use of OLS
regression in its multiple form, Yon multiple X, is familiar to
most remote sensing analysts conducting regression modeling. Although much less familiar, there is also a formulation
for nmltiple X inverse calibration (Brown. 1979). A simple
application of RMA requires one X and one Y Thus, to
incorporate n-space indices and/or temporal datasets into an
RMA, the multiple Xdataset must be linearly combined into
a single X variable. In essence, we must develop a new,
integrated index that is a linear combination of the multiple
X indices (or bands) from a single date or multiple dates. As
discussed earlier, this need is directly facilitated by CCA.
CCA is a generalized form of multiple regression that
permits the examination of interrelationships between two
sets of variables (multiple X's and multiple Y's) (Tabachnick
& Fidell, 1989). CCA maximizes the correlation between a
composite of variables from one set with a composite of
variables from another set. When there is only one X (i.e.,
vegetation variable, such as LAI), CCA provides a set of
coefficients for the Y's that aligns them with the variation in
the X variable. When those coefficients are applied to the Y
variables, the result is a CCA score for each observation.
CCA scores are indexed values in the same way that
brightness, greenness, or wetness (or NDVI) values are
indexed values. However, with CCA, the alignment is
dataset-specific, whereas with the Tasseled Cap or NDVI,
the formulations are generalized and fixed.
2. Methods
This work was conducted in a temperate broadleaf agroecosystem, consisting of corn and soybeans, and a boreal
needleleaf evergreen forest. The biophysical variable of
interest within the agro-ecosystem was LAI, which was
modeled for two separate measurement dates. The dominant
tree species in the boreal forest is black spruce, and the
variable we modeled was percent tree cover. This work was
done in the context of the BigFoot project, which was
designed to provide local validation of global estimates of
biophysical variables and processes using MODIS data
(Cohen & Justice, 1999).
2.1. Study sites, sampling design, and field measurements
The study sites and sampling design were described in
Campbell, Bun'ows, Gower, and Cohen (1999). The agricultural site (AGRO) was a 5 × 5 km area located just south
of Champaign, IL. The boreal forest site was a similarly
sized area surrounding the northern old black spruce
(NOBS) site of the Boreal Ecosystem Atmosphere Study
(Sellers et al., 1997), approximately 40 km west of Thompson, Manitoba, Canada. The sample design was a nested
spatial series (Burrows et al., in press) that permits explicit
examination of spatial covariation among field-measured
ecosystem properties using variograms and cross-variograms (Cressie, 1991). At each site, there were approximately 100 25 × 25 m plots where land cover, LAI,
absorbed radiation, and net primary production were measured/observed at five to nine subplots per plot. Subplot
measurements were averaged to provide a single value for
each measured variable at each plot. Plot locations were
determined using a real-time differential GPS. The accuracy
of the system was < 0.5 m in both the x and y dimensions.
At the AGRO site, LAI was measured at five subplots per
plot using standard, direct harvest methods described by
Gower et al. (1999). Measurements were made at several
time periods during the growing season in 2000. We used
data from July and August. At NOBS, percent tree cover
was measured at nine systematically spaced subplots using
an upward-looking digital camera. The imaged canopy
projection area was dependent on tree height and the field
of view of the camera, which was 30 °. At approximately 10m height, this means that among the nine subplots, nearly
W.B. Cohen et aL / Remote Sensing of Environment 84 (2003) 561-571
I00% of the canopy area in each plot was imaged. In the
lab, each of the nine photos per plot was sampled using a
grid of 99 points to derive the percent live tree canopy cover
at each plot (Berterretche, 2002).
2.2. Image data and processing
For AGRO, ETM+ data from four dates were used to
capture the growing season from April through September
(1-able I). At NOBS, two images were used, one from
March and one from June. The images were georeferenced,
radi.ometrically calibrated, and translated into Tasseled Cap
brightness, greenness, and wetness. All images were acquired at level 1G processing, with a cell size of 30 m, and
UTM (WGS84) projection. At AGRO, positional accuracy
of the native map projection of the June image was judged
by direct comparison with USGS digital orthophoto quadrangles (DOQs) at a 9 x 9 km area centered on the study
site. A systematic local shift of - 37.5 m in the x-direction
and - 127.5 m in the y-direction was applied to the ETM+
image to register it to the DOQs. Subsequently, all other
image dates were positionally shifted to match the June date.
At NOBS, a panchromatic IKONOS image was registered to
the earth's surface with the same projection parameters as at
AGRO using several GPS points collected in the field. The
June image was then positionally shifted to match the
IKONOS image, and the March image was shifted to match
the June image.
The COST absolute radiometric correction model of
C,havez (1996) was applied to each image to convert digital
counts to reflectance. Radiometrically "dark" objects were
assumed to have 2% reflectance across all bands. For
AGRO, the June image was selected as a reference image
and all other dates of imagery were relatively normalized to
it, as a fine-tuning for multidate, inter-image calibration.
The method used was similar to that of Oetter et al. (2001)
and of the Ridge Method of Kennedy and Cohen referred to
by Song, Woodcock, Seto. Pax I.enney. and Macomber
(2001), which are an adaptation of standard band-by-band
relative normalization procedures based on co-located bright
and dark targets. As the COST model is not appropriate for
low sun angle situations, the March image from NOBS was
converted to reflectance using a more basic dark-objectsubtraction model. Further, no relative normalization was
performed for the NOBS dataset due to major spectral
property differences between the two dates, given the backTable 1
ETM+ images used in this study
Site
Path/row
Date
AGRO
22/32
22/32
22/32
22/32
34/21
33/21
April 26, 2000
June 29, 2000
July 15, 2000
September 1, 2000
March 13, 2000
June 6, 2000
NOBS
565
drop of ice and snow for the March image and of vegetation
and water for the June image.
No published transformation exists to convert atmospherically-corrected ETM+ spectral data to Tasseled Cap
indices. However, Crist (1985) derived coefficients for
brightness, greenness, and wetness from ground-based
spectral data that can be applied to atmospherically corrected Lands~it data. Slight differences in spectral band
width and position, as well as calibration, exist between
Landsat TM and ETM+ (Teillel el al., 2001; ~bgelmann et
al., 2001), but they are similar enough to assume that the
differences in Tasseled Cap indices derived for data from the
two different sensors are small. We tested this assumption
using TM and ETM+ images acquired within a few days of
each other (Path 46/Row 29) over western Oregon in 1999.
First, we converted atmospherically corrected TM DN data
to the Tasseled Cap indices using the coefficients in Crist
and Cicone (I 984). We then converted the atmospherically
corrected TM DN data to reflectance using published
coefficients and formulae, before using the Crist (1985)
coefficients to convert the reflectance data to Tasseled Cap
indices. Finally, we atmospherically corrected the ETM+
data and then converted the reflectance data to the Tasseled
Cap indices using the Crist (1985) coefficients. A comparison of the brightness, greenness, and wetness images from
the three methods showed that they were highly intereorrelated at a level of roughly 95%.
2.3. Variable selection and model development, execution,
and comparison
With both OLS and RMA regression done in a multipleY (i.e., multiple SVIs) context, there is the issue of variable
selection. Not all Yvariables are needed or are significant in
the presence of other Yvariables, and some may need to be
culled from the dataset. For this we used forward stepwise
regression. In each case, brightness, greenness, and wetness
from all dates of available imagery were used as potential
variables for a model. To avoid overfitting a given model,
we imposed the rule that the number of variables to enter the
model be less than one-third the number of observations.
Prior to conducting stepwise regression, bivariate plots of all
potential Y variables against LAI or canopy cover were
evaluated to determine if transformations were required to
linearize relationships. Where necessary, standard log and
square root transformations were used.
Once the variables for a given dataset were selected,
RegT, Regl, and RMA regression models were developed.
For all three modeling approaches, tl~e CCA axis derived
from the same Y-variable set was used. To compare the three
modeling approaches, predicted versus observed plots were
developed and overall bias and variance ratios were calculated. Bias was calculated as the mean of the predicted
values minus the mean of the observed values, such that a
positive bias equated to a mean overprediction and vice
versa. Variance ratio was calculated as the standard devia-
I~B. Cohen et al. / Remote Sensing of Environment 84 (2003) 561 571
566
50
45
o
i
Oo:
40
35
~ 30
........
............
! .....
.
e-
:
.........
.
i--'-'-++--"
:
--
.... -+u----u~.. . . . . . . . . . . . ". . . . . . . . . . . :@- . . . . . . . . . ". . . . . . . . . . . : . . . . . . . . . . . .
~ 25
__~ 20
-',1
~
estimator of prediction error (Efron & Gong. 1983). This
required, for each dataset and regression variant, that (where
n = 100) 100 separate models be developed, each time with
data from 99 observations. Then, each model was used to
predict the observation that was left out, thus providing the
predictions for all 100 plot observations that were needed to
compare against the observed values. This provided an error
characterization equivalent to the PRESS statistic (SAS,
1990).
0
15
. . . . 0 . . . . . . . , . . . . . . . . . . . t. . . . . . . . . . . . . . .
;+ . . . . . . . . . . . . . . . . . . . . . . . .
10 ........... :............ ; ........... ~........ • Corn
.......
............ +............ +----....... i........ o Soybeans .......
0
I
I
i
I
I
1
2
3
4
5
July LAI
3. Agricultural e x a m p l e - - L A l at AGRO
A scatterplot of corn and soybean greenness from the
July measurement date (Fig. 1) revealed that these two crops
represented different populations and were best modeled
separately. This was done for both July and August dates,
yielding four separate modeling sets (Table 2). For all four
model sets, the three regression approaches had equivalent
Fig. 1. Tasseled Cap greenness as a function o f July LAI at AGRO.
tion of the predicted values divided by the standard deviation of the observed values. As such, a ratio of greater than
one meant that the prediction variance was greater than the
observed variance.
• Field data are expensive to collect and process, so using
them prudently is essential. There is a trade-off between
using all available observations to develop a regression
model and having no independent observations to test the
model, versus excluding a predetermined number of observations to test the model, but having a less robust model
because it was developed on fewer points. The statistical
literature provides several alternative, but related ways to
address this problem: cross-validation, bootstrapping, and
jackknifing CEfron & Gong, 1983~. We used the crossvalidation procedure, which provides a nearly unbiased
Table 2
Regression model statistics for each model type and dataset
Model
Type
Slope
Intercept
R2
Soy, July
Regr
Regt
RMA
Regv
Regl
RMA
Regv
Regt
RMA
Regr
Regt
RMA
Regv
Regl
RMA
0.49
1.19/0.84
0.64
0.63
0.96/1.04
0.81
0.49
0.56/I .79
0.93
0.45
1.44/0.70
0.56
15.08
0.045/22.1
18.26
1.54
- 1.83/1.54
1.54
4.41
- 4.24/4.41
4.41
3.44
- 1.91/3.44
3.44
4.00
- 5.76/4.00
4.00
38.89
- 1.76/38.89
38.89
0.58
0.58
0.58
0.61
0.61
0.61
0.27
0.27
0.27
0.64
0.64
0.64
0.68
0.68
0.68
Corn, July
Soy, August
Corn, August
Canopy cover
For the Reg~ models, the original slope and intercept are given along with
the back-inverted slope and intercept for comparison with other model
types.
Table 3
Cross-validation results for each model type and dataset
Model type
Soy, July
R
RMSE
Bias
Variance ratio
Corn, July
R
RMSE
Bias
Variance ratio
Combined, July
R
RMSE
Bias
Variance ratio
Soy, August
R
RMSE
Bias
Variance ratio
Corn, August
R
RMSE
Bias
Variance ratio
Combined, August
R
RMSE
Bias
Variance ratio
Canopy cover
R
RMSE
Bias
Variance ratio
n
Regv
Regl
RMA
0.74
0.42
- 0.01
0.80
0.74
0.59
- 0.03
1.40
0.74
0.47
- 0.01
1.06
0.76
0.53
0.00
0.76
0.76
0.68
0.03
1.31
0.76
0.56
0.01
1.01
0.95
0.46
- 0.01
0.96
0.92
0.62
- 0.01
1.07
0.94
0.50
0.00
1.00
0.47
0.82
- 0.01
0.53
0.49
1.57
- 0.02
1.95
0.49
0.95
- 0.02
1.02
0.79
0.35
0.00
0.81
0.79
0.46
0.03
1.33
0.79
0.37
0.01
1.02
0.59
0.70
- 0.01
0.62
0.54
1.32
- 0.01
1.81
0.57
0.81
- 0.01
1.01
0.82
10.41
0.02
0.83
0.82
12.68
0.03
1.22
0.82
10.93
0.03
1.01
64
31
95
64
31
95
103
For the agricultural site, five plots were not corn or soybeans• For the forest
site, there were three extra plots.
567
W.B. Cohen et al. /Remote Sensing of Environment 84 (2003) 561-571
coefficients of determination and model intercepts. The
differences among approaches were expressed in the slope
term, with Regm, having the least, Regl having the greatest,
and RMA being intermediate. As mentioned earlier, these
are anticipated results that are provided here for demonstration purposes.
A summary of cross-validation predictions revealed the
effect of the different modeling approaches (Table 3), again,
fbr demonstration purposes. Presented are results from the
crop-specific models and from the combined set of predictions across the two crop-specific models. For the July date,
corn. soybeans, and combined, the correlation coefficients
(R) between predicted LAI and observed LAI were essentially the same for all three approaches. The only difference,
which was minimal, was for the combined model. This
difference is attributable to the cross-validation procedure.
Bias was near zero in all cases, indicating that the observed
mean of the samples was preserved in the predictions. The
differences among the modeling approaches were related to
the different slope terms (from Table 2), and were expressed
in both the RMSE and the variance ratio. RegT, by design,
had the lowest RMSE in predictions of Y (LAI). Similarly,
because Regl also minimized the sums of squares of error in
Y (this time, SVI), it yielded the greatest RMSE in LAI. As
RegT
,
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Fig. 2. Predicted (from cross-validation)versus observedJuly and August LAI at AGRO. RegTis traditionalOLS regression. Regl is inverse OLS regression,
and RMA is reduced major axis regression. Left is July; right is August.
568
W.B. Cohen et al. /Remote Sensing of Environment 84 (2003) 561-57l
expected, RMA was a compromise solution, having intermediate values of RMSE. With respect to the variance ratio,
RMA always exhibited a value close to 1.0, indicating that
the variance structure of the observed values was preserved
RegT
100
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4. B o r e a l f o r e s t e x a m p l e - - t r e e
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Reg
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Observed
in the predicted values. Deviations from unity for the OLS
methods were greatest when the correlation coefficients
were lowest. This latter point was particularly evident in
the results from August, where correlations were lower than
in July. For all cases, Reg~ had variance ratios greater than
1.0 and RegT had values less than 1.0, indicating greater and
lesser variance, respectively, relative to observed values.
With regression models, it is always possible to have
individual predictions outside the range of observed values.
This mostly occurs when observed "independent" variables
have values outside the range on which models were
constructed (an increased possibility with cross-validation),
or when significant outliers exist in the model dataset. Here,
this is evident from the predictions of negative LAI (Fig. 2).
Of course, this "problem" is amplified with Regl and is
suppressed with RegT.
t
I
100
Cover
RMA
c o v e r at N O B S
The models for tree cover at NOBS exhibited similar
relative characteristics as those at AGRO (Table 2). The
coefficients of determination and the intercepts were all
identical. The only difference among modeling approaches
was the slope of the relationships, with Regv having the
lesser value, Regl having the greater value, and RMA
having an intermediate value. Likewise, the cross-validation
results indicated identical correlations between tree cover
and the CCA axis and essentially no overall bias for any of
the models. Again, RMSE was lowest for Regv, highest for
Reg[, and intermediate for RMA.
Some predictions outside of the observed range occurred
for tree cover, as it had for LAI at AGRO (Fig. 3). Again, this
was most evident using the Regl approach, which amplified
the variance of the predictions relative to the observed values.
For tree cover, there was a slight tendency toward an
asymptote in the predictions, especially for the Regx model.
100
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t
i
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40
60
80
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- -
100
Cover
Fig. 3. Predicted (from cross-validation)versus observedpercent live tree
canopycover at NOBS. Regvis traditional OLS regression, Regl is inverse
OLS regression, and RMA is reduced major axis regression.
This paper presents to a remote sensing audience a
verification of existing regression theory. The remote sensing literature contains little of regression theory, and even
less of the numerous options for its application. With rare
exception, the remote sensing literature contains rote application of OLS (ordinary least squares) regression, without
ever questioning its disadvantages relative to other forms of
regression. Questioning these disadvantages and demonstrating alternative regression approaches were the main
purposes of this paper. Numerous examples of alternative
regression models exist in many other scientific and technical fields, but remote sensing community has largely
ignored the important work of Curran and Hay (1986).
Most of the regression-based remote sensing literature is
focused on minimizing error (e.g., RMSE) in the predictions
~B. Cohen et al. /Remote Set,sing of Environment 84 (2003) 561-571
of biophysical variables. Traditional OLS regression is an
excellent means for accomplishing that, but there are two big
problems with OLS. First, it assumes no error in measurements of vegetation reflectance and/or the biophysical variable of interest. Second, traditional OLS provides attenuated
variance in predictions of that variable. The statistical
literature strongly suggests that if there are errors in the
measurements of both variables (e.g., reflectance and biophysical), then OLS regression is the 'wrong model to use.
Because it is nearly impossible to defend any claim that
either reflectance or biophysical variables are measured
without error, application of OLS regression is inappropriate
in remote sensing.
Compression of variance by OLS becomes critical if the
regression model is used to build a map of a biophysical
variable, that in turn drives a functional/mechanistic model.
If the mechanistic model involves nonlinear functions of the
biophysical variable, attenuation of variance in the biophysical variable introduces error in the mechanistically modeled
outputs. Because OLS attenuates the variance, it will introduce such error. The degree of attenuation is essentially a
linear function of the correlation between the spectral data
and the biophysical variable, low correlation, much attenuation, and vice versa. Many of the relationships between
reflectance and biophysical variable are poorly correlated, so
this is not a non-issue.
The remote sensing literature contains a great number of
examples of empirical, regression modeling relating SVIs to
measures of a myriad of vegetation variables modeled across
an assortment of sites and biomes. Most studies used SVIs
based on red and near-infrared reflectance. More often than
not, a single SVI from a single date was used. The spectral
depth of ETM+ is essentially three-dimensional (Cohen et
al.. 1995: Crist & Cicone, 1984) and we have increasing
numbers of temporal image series available for greater
predictive power (I_el%ky el al., 2001 ). As such, we should
be expanding our use of multiple regression over simple
regression techniques. Multiple regression'in an RMA context requires a single-integrated index of multiple bands or
indices. This need is directly facilitated by CCA. Canonical
correlation analysis has rarely been used in remote sensing.
However, tbr those contemplating the use of CCA for
deriving a dataset-dependent index, there should be a clear
understanding of what the procedure does to the dataset.
A simple test on any appropriate single-Y (in this case,
e.g., LAI), multiple-X (in this case, SVls) dataset illustrates
that CCA scores are perfectly correlated to predicted Yvalues
from traditional OLS multiple regression on that dataset. The
difference is that one provides predictions of the Yvariable,
whereas the other is simply a set of index scores that are
maximally correlated with the observed Y variable. If one
then conducts traditional simple OLS regression with the
CCA scores as X and the LAIs as Y, they will derive exactly
the same predicted values for LAI as those predicted from
the original multiple OLS regression. In both cases, RMA
would be required to balance the variance ratio at a value of
569
1.0. Traditional OLS regression provides biophysical predictions, but the variance of those predictions is unbalanced
vis-/l-vis the observed variance in the biophysical variable.
CCA provides an index that is maximally correlated with the
biophysical variable of interest, but it does not provide
predictions. RMA can either provide predictions from a
CCA index that have a balanced variance, or it can balance
(or calibrate) a set of unbalanced predictions derived previously from traditional OLS regression conducted on a
CCA index or from multiple OLS regression. For the latter
case, the CCA index would be superfluous. Thus, the only
important reason for conducting the CCA is if the index itself
is desired, for which there may be numerous reasons. For this
study, it was desirable to have a single index for the
convenience of comparative analysis among methods to
derive a single regression slope term for each method. It is
important to keep in mind that whereas CCA is datasetspecific, NDVI or SR, or other SVIs such as brightness,
greenness, and wetness, are more generalizable in terms of
their biophysical meaning.
In this study, we illustrated an improved regression
modeling strategy that incorporates all the recommended
steps for deriving mapped estimates that have their errors
characterized. This strategy includes the collection of georeferenced field data, image georeferencing, image radiometric
calibration, translation of reflectance into SVIs, testing for
significance of each SVI in regression models, and using
cross-validation to provide nearly unbiased testing of robust
models. Applying the RMA models to the CCA indices
provided high-quality maps of LAI for the agricultural site,
with means and variances well preserved in each important
land cover class (corn and soybeans). Additionally, an
important forest variable was mapped, tree cover, which will
subsequently be used at the forest site to help derive a land
cover map using classes that are largely based on percent tree
cover. Although any given study may weigh these various
processing components differently, two considerations are
critical. (1) Is it acceptable for a predicted variable to have a
different variance structure from that of empirical observations? (2) Is there a compelling reason to limit the analysis to
a single SVI from a single date? If the answer to Question 2
is "no", then CCA may be an important aid in your analysis.
Acknowledgements
This research was funded by NASA's Terrestrial Ecology
Program. We greatly thank Karin Fassnacht for thoughtful
and lively discussion and for her review of early drafts, and
Robert Kennedy for his contributions to the discussion.
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